Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Billy gets paid at home for doing extra chores. Last week, he did $4$ loads of laundry and $2$ loads of dishes, and his parents paid him $\$10$ . The week before, he finished $6$ loads of laundry and $7$ loads of dishes, earning a total of $\$19$ . How much does Billy earn for completing each type of chore? $\newline$ Billy earns $\$\_\_\_\_$ per load of laundry and $\$\_\_\_\_$ per load of dishes.

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Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Billy gets paid at home for doing extra chores. Last week, he did

4

loads of laundry and

2

loads of dishes, and his parents paid him

\$10

. The week before, he finished

6

loads of laundry and

7

loads of dishes, earning a total of

\$19

. How much does Billy earn for completing each type of chore?

\newline

Billy earns

\$\_\_\_\_

per load of laundry and

\$\_\_\_\_

per load of dishes.

Define earnings per task: Let's denote the amount Billy earns per load of laundry as $l$ and per load of dishes as $d$ . From the first week, we have the equation $4l + 2d = 10$ . This represents the $\(10$)$ he earned from $$4$$ loads of laundry and $$2$ loads of dishes.
Formulate system of equations: From the week before, we have the equation $6l + 7d = 19$ , which represents the $\$19$ he earned from $6$ loads of laundry and $7$ loads of dishes.
Use elimination method: We now have a system of two equations: $\newline$ $1$ . $4l + 2d = 10$ $\newline$ $2$ . $6l + 7d = 19$ $\newline$ We will use elimination to solve this system. To eliminate one of the variables, we can multiply the first equation by $3$ and the second equation by $2$ , so the coefficients of $l$ will be the same.
Solve for cost of dishes: Multiplying the first equation by $3$ gives us $12l + 6d = 30$ . Multiplying the second equation by $2$ gives us $12l + 14d = 38$ .
Substitute and solve for laundry cost: We now subtract the first new equation from the second new equation to eliminate $l$ . This gives us $12l + 14d - (12l + 6d) = 38 - 30$ , which simplifies to $8d = 8$ .
Substitute and solve for laundry cost: We now subtract the first new equation from the second new equation to eliminate $l$ . This gives us $12l + 14d - (12l + 6d) = 38 - 30$ , which simplifies to $8d = 8$ .Dividing both sides of $8d = 8$ by $8$ gives us $d = 1$ . This means Billy earns $\$1$ per load of dishes.
Substitute and solve for laundry cost: We now subtract the first new equation from the second new equation to eliminate $l$ . This gives us $12l + 14d - (12l + 6d) = 38 - 30$ , which simplifies to $8d = 8$ .Dividing both sides of $8d = 8$ by $8$ gives us $d = 1$ . This means Billy earns $\$1$ per load of dishes.We substitute $d = 1$ into the first original equation $4l + 2d = 10$ and solve for $l$ . This gives us $12l + 14d - (12l + 6d) = 38 - 30$ $0$ , which simplifies to $12l + 14d - (12l + 6d) = 38 - 30$ $1$ .
Substitute and solve for laundry cost: We now subtract the first new equation from the second new equation to eliminate $l$ . This gives us $12l + 14d - (12l + 6d) = 38 - 30$ , which simplifies to $8d = 8$ .Dividing both sides of $8d = 8$ by $8$ gives us $d = 1$ . This means Billy earns $\$1$ per load of dishes.We substitute $d = 1$ into the first original equation $4l + 2d = 10$ and solve for $l$ . This gives us $12l + 14d - (12l + 6d) = 38 - 30$ $0$ , which simplifies to $12l + 14d - (12l + 6d) = 38 - 30$ $1$ .Subtracting $12l + 14d - (12l + 6d) = 38 - 30$ $2$ from both sides of $12l + 14d - (12l + 6d) = 38 - 30$ $1$ gives us $12l + 14d - (12l + 6d) = 38 - 30$ $4$ . Dividing both sides by $12l + 14d - (12l + 6d) = 38 - 30$ $5$ gives us $12l + 14d - (12l + 6d) = 38 - 30$ $6$ . This means Billy earns $12l + 14d - (12l + 6d) = 38 - 30$ $7$ per load of laundry.